Support-vector leave-one-out error bound (${\mathrm{err}}_{CV}\le s/n$)

For a fixed $C$ in the unnormalized sum-of-slacks objective and unique training optimizers, deleting an observation with ${\alpha }_i=0$ preserves the Karush-Kuhn-Tucker conditions and the fitted decision function. It is correctly classified when held out. Consequently Leave-one-out cross-validation makes at most $s$ errors, where $s$ counts positive dual coefficients. With geometric support vectors, counting all points of signed margin at most one also gives the bound, without a nondegeneracy assumption.